Arcsine and Darling–Kac laws for a piecewise linear random interval map

The paper proves arcsine and Darling–Kac laws for the Hata random map via Markov partitions, conjugacy to a Markov chain on the λ-partition, wandering-rate asymptotics, and the Thaler–Zweimüller framework, obtaining exactly the two limit laws in Theorem 1.2. The candidate solution reaches the same two conclusions by a different, renewal/local-time route: it coarse-grains the dynamics into side/depth variables, derives heavy-tailed sojourns, invokes Lamperti’s arcsine law, and identifies the √N-normalized visit counts with Brownian local time. One technical imprecision in the model solution is the identification of a side-sojourn length with a single hitting time from a geometric height; in fact it is a geometric sum of such hitting times, but this does not affect the limiting laws. Overall, both are correct; the paper’s proof is rigorous and complete; the model’s proof outline is essentially correct but glosses some details.